Each of these words refers to anything that is suitable for a king or closely associated with one: "The tall old man with a long white beard had a kingly (or regal or royal) look about him." Each also has a meaning of "stately, splendid": "The ambassador and his staff made a regal (or royal or kingly) entrance into the ballroom." Kingly is a word closely associated with a monarch (a kingly crown); regal refers to the grandeur and majesty of the office of kingship (regal authority, regal robes); royal applies particularly to the person (body) of the king (royal residence, royal bedchamber).
Dictionary of problem words and expressions. Harry Shaw. 1975.